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Axiomatic cellular boundaries are integral incidence matrices with coefficients

Statement

For a CW pair $(X, A)$ , an ordinary theory $h$ with coefficient $G$ , and chosen cell orientations, the complex $C_{*}^{h} (X, A)$ is canonically $C_{*}^{c e l l} (X, A; Z) \otimes G .$ Its differential is the integral incidence matrix acting on $G$ . In degree one the entries are signed terminal-minus-initial endpoints. The direct-sum matrices have finite support in each column.

Facts & Assumptions

Given: The objects and hypotheses in the statement above.

[F1]

For a CW pair $(X, A)$ and an ordinary theory $h$ with coefficient $G$ , put $F_{- 1} = A$ and $F_{r} = A \cup X^{r}$ for $r \geq 0$ . Set $C_{r}^{h} (X, A) = h_{r} (F_{r}, F_{r - 1}) (r \geq 0), C_{r}^{h} = 0 (r < 0) .$ Each $C_{r}^{h}$ is the direct sum of copies of $G$ indexed by the relative $r$ -cells. Define $d_{0} = 0$ and for $r \geq 1$ let $d_{r}$ be the triple boundary to $h_{r - 1} (F_{r - 1}, A)$ followed by its map to $h_{r - 1} (F_{r - 1}, F_{r - 2})$ . Then $d_{r - 1} d_{r} = 0$ , naturally for cellular maps of CW pairs. (Any ordinary homology theory has a cellular chain complex on a cw pair)

[F2]

Let $n \geq 0$ and $u : S^{n} \to S^{n}$ be continuous. If $u_{*}$ on ${\tilde{H}}_{n} (S^{n}; Z)$ is multiplication by $d$ , then $u_{*}$ on ${\tilde{h}}_{n} (S^{n}) ≅ G$ for every ordinary theory $h$ is $d {i d}_{G}$ . The identifications use the same oriented sphere generator; for $n = 0$ use the difference of the two point classes. (Sphere endomorphisms act by the same integer in every ordinary theory)

[F3]

For $n \geq 2$ and oriented cells $e_{α}^{n}$ and $e_{β}^{n - 1}$ , collapse the complement of $e_{β}^{n - 1}$ in $X^{n - 1}$ and compose the attaching map of $e_{α}^{n}$ with the resulting quotient to $S^{n - 1}$ . Its induced endomorphism of oriented ${\tilde{H}}_{n - 1} (S^{n - 1}; Z)$ is multiplication by a unique integer, denoted $[e_{α}^{n} : e_{β}^{n - 1}]$ . For $n = 1$ , choose the characteristic interval compatibly with the orientation of $e_{α}^{1}$ , running from $- 1$ to $+ 1$ . Write the chosen vertex generator as $e_{v} = ε_{v} [v]$ , where $[v]$ is the canonical positive point class and $ε_{v} \in {1, - 1}$ . Then $[e_{α}^{1} : e_{v}] = ε_{v} (1_{{χ_{α} (+ 1) = v}} - 1_{{χ_{α} (- 1) = v}}) .$ For canonical positive vertex generators this is terminal minus initial; a loop still has incidence number zero at its common endpoint. (Incidence number of two CW cells)

[F4]

Let $X$ be a CW complex. For $n \geq 1$ , in the integral cellular chain groups with the chosen cell orientations, $d_{n} e_{α}^{n} = \sum_{β} [e_{α}^{n} : e_{β}^{n - 1}] e_{β}^{n - 1} .$ (Cellular boundary is the incidence degree matrix)

Proof

1.1F1F3F4

By F1 the chain groups are direct sums of copies of $G$ on relative cells. For a source $r$ -cell and target $(r - 1)$ -cell with $r \geq 2$ , project the boundary homomorphism onto the target summand. Naturality of the characteristic disk pair identifies this component with the attaching map followed by collapse onto the target cell sphere and the natural disk boundary identifications. The corresponding integer for integral singular homology is precisely the incidence number of F3 and F4.

2.1F2F3step 1.1

F2 says that this same sphere endomorphism acts on coefficient $G$ by that integer times the identity. For $r = 1$ , the boundary of the compatibly oriented interval is $(- g, g)$ in the canonical point coordinates at its initial and terminal ends. In the target coordinate belonging to $e_{v} = ε_{v} [v]$ , the coefficient of $[v] \otimes g$ is $ε_{v} g$ , since $[v] = ε_{v} e_{v}$ . Thus projection to the $v$ -summand gives $ε_{v} (1_{{v_{+} = v}} - 1_{{v_{-} = v}}) {i d}_{G}$ , precisely the incidence entry of F3. With every $ε_{v} = 1$ the old terminal- $g$ minus initial- $g$ formula is recovered. Coincident endpoints cancel, and endpoints in $A$ vanish in the relative complex. A negative vertex generator changes the algebraic coefficient coordinate by $- {i d}_{G}$ ; it is not implemented by a based degree- $- 1$ self-map of $S^{0}$ .

3.1F1step 1.1step 2.1∎

Each characteristic boundary has image in the finite union of closed cells supplied by closure finiteness, so only finitely many target cells can contribute to its column. Thus these components define a map of direct sums. At degree zero the outgoing differential is zero. The bases and component calculations identify the entire complex with the displayed tensor complex, for arbitrary $G$ , including $G = 0$ and pairs with no relative cells.

Depends on

Used by

Finite additivity alone does not prove infinite cw uniqueness Counterexample
Steenrod squares do not all lift integrally Counterexample
Top squares do not determine lower squares Counterexample
Steenrod squares on real projective space Example
Finite dimensional axiomatic homology has finite subcomplex support Lemma

Dependency tree · two levels

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Sources

May, A Concise Course in Algebraic Topology, 15§2, first theorem and arbitrary-coefficient paragraph p.119 (standard reference, not scraped)